Bohr Model
Atomic Radius
The Bohr Model is a semiclassical model for hydrogen atoms. It uses concepts from classical mechanics, such as the centripetal force, but additionally a quantized angular momentum.
Similar to the movements of the planets in our solar system, Nils Bohr assumed that the electron inside the hydrogen atom orbits around the proton. In order to find the radius, one can first equal the centripetal and Columb force:
$$\frac{m_ev^2}{r} = \frac{e^2}{4\pi\varepsilon_0r^2}$$
Solving this equation for $r$ results in
$$r = \frac{e^2}{m_ev^24\pi\varepsilon_0 r^2}$$
Inserting the quantized angular momentum leads to the calculation of the radii of the different energy levels.
$H$ Atom Radius
The radius of the electron inside a hydrogen atom is given for each energy level $n$ as
$$r_n = n^2 \frac{4\pi\varepsilon_0\hbar^2}{m_ee^2}$$
where $m_e$ is the elctron mass and $e$ the elementary charge.
Energy Levels
In the next step, we can multiply the first equation simply with r. The left side can then be identified with the double kinetic energy, whereas the right side equals the potential energy of a Columb field:
$$E_\mathrm{kin} = \frac{1}{2}E_\mathrm{pot}$$
This correlation is called the
virial theorem. The total energy of the electron is then given as
$$E = E_\mathrm{kin} + E_\mathrm{pot}$$
which then leads to
$$E = \frac{1}{2}E_\mathrm{pot} = -\frac{1}{2}\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r}$$
Inserting the previously defined condition for the radius, this equation can be rewritten as follows.
Energy Levels
The potential energy of an electron can be calculated according to
$$E_n = -\left(\frac{e^2}{4\pi\varepsilon_0}\right)^2\frac{m_e}{2\hbar^2}\frac{1}{n^2}$$
This formula can be used to calculate the energy levels of a hydrogen atom. The numerical value depends only on natural constants and the shell of the electron.
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Last modified: 2022-10-06 16:35:17 by mustafa
